I am trying to prove that if $C$ is a connected component of $X$, then $C$ is closed. Here is my attempt:
Let $C$ be a connected component of $X$. Then $\overline{C} \supseteq C$ must be connected as well, and since every connected subspace intersects one, and only one, connected component (and is therefore contained in it), it follows that $\overline{C} \subseteq C$.
Is this right? Something about it is fishy…
Best Answer
A component of $x$ is the largest connected set containing $x$.
Let $C$ be a component of $x$. Thus $x\in\overline{C}$ and $C \subseteq \overline{C}$.
However, $C$ is the largest connected set, therefore $\overline{C} \subseteq C$.
Hence $C = \overline{C}$, and $C$ is closed.